Background: The cluster radioactivity from the neutron-deficient trans-tin region of the nuclear landscape has given immediate attention in the nuclear structure studies. Recent prediction of the emitting clusters from the ground and intrinsic excited states of proton-rich Ba isotopes opens the direction to explore the corresponding decay characteristics. A theoretical probe is necessary for understanding the cluster decays of Ba isotopes.

The newly derived relativistic energy density functional [ Phys. Rev. C 103, 024305 (2021)], which stems from the effective field theory motivated relativistic mean field (E-RMF), is employed to establish the appearance of peaks and/or kinks in the symmetry energy over the isotopic chain of Pb nuclei. The coherent density fluctuation model parametrization procedure for finite nuclei is adopted here to obtain the relativistic energy density functional at local density. The relativistic energy density functional from relativistic mean-field theory takes precedence over the Brückner energy density functional as it accurately predicts the empirical saturation density and binding energy per nucleon $E/A$, so-called coester band problem. Interestingly, using the relativistic energy density functional, it is possible to predict the peak at $N=126$ for recently developed G3 and widely used NL3 parameter sets, which is not observed for Brückener's functional in spite of using the relativistic mean-field density. From the present analysis, the newly fitted energy density functional is found to be minutely sensitive to the choice of the parameter sets employed.

A new $\alpha$-emitting ²¹⁴U has been recently observed experimentally. This opens the window to theoretically investigate the ground-state properties of the lightest known even–even neutron deficient ^214,216,218U isotopes and to examine $\alpha$-particle clustering around the shell closure. The decay half-lives are calculated within the preformed cluster-decay model (PCM). To obtain the $\alpha$-daughter interaction potential, the RMF densities are folded with the newly developed R3Y and the well-known M3Y NN potentials for comparison. The alpha preformation probability $\left(P_{\alpha }\right)$ is calculated from the analytic formula of Deng and Zhang. The WKB approximation is employed for the calculation of the transmission probability. The individual binding energies (BE) for the participating nuclei are estimated from the relativistic mean-field (RMF) formalism and those from the finite range droplet model (FRDM) as well as WS3 mass tables. In addition to $Z=84$, the so-called abnormal enhancement region, i.e., $84\le Z\le 90$ and $N<126$, is normalised by an appropriately fitted neck-parameter $\Delta R$. On the other hand, the discrepancy sets in due to the shell effect at (and around) the proton magic number $Z=82$ and 84, and thus a higher scaling factor ranging from ${10}^{-8}$–${10}^{-5}$ is required. Additionally, in contrast with the experimental binding energy data, large deviations of about 5–10 MeV are evident in the RMF formalism despite the use of different parameter sets. An accurate prediction of $\alpha$-decay half-lives requires a Q-value that is in proximity with the experimental data. In addition, other microscopic frameworks besides RMF could be more reliable for the mass region under study. $\alpha$-particle clustering is largely influenced by the shell effect.

This study presents a holistic picture of the preformation of nuclear clusters with credence to the kinematics of their emissions. Besides the fitting of the preformation formula to reproduce the experimental half-lives, we have investigated the interrelationship between the parameters involved in the cluster decay process for medium, heavy and superheavy nuclei. Based on the established conceptual findings, we propose a new cluster preformation probability (P ₀) formula that incorporates all influential parameters of the cluster radioactivity and thus has an edge over the existing formulae in the literature. Further, we hypothesize that a fraction of the decay energy is needed for cluster formation within the parent nucleus. The proposed formula opens a new paradigm to separately estimate the energy contributed during the cluster formation from its emission and thus shows that the contribution of the Q-value splits into three major parts accounting for the energy contributed during the cluster preformation, its emission and recoil of the daughter nucleus. Moreover, the expression P ₀ is adept at accommodating the theorized concept of heavy particle radioactivity (HPR). The result reveals that, like α-decay, a proper estimation of the P ₀- and Q-value in the cluster studies is enriched with qualitative information about the nuclear structure. However, from the analysis, the Geiger-Nuttall law is not the best compromise in the clustering due to the non-linearity between log₁₀ T _1/2 and , unlike in α-decay. We have demonstrated that with the inclusion of the proposed formula, the half-life predictions from both microscopic R3Y and phenomenological M3Y NN potentials closely agree with the available experimental data and that the slight variation can be traced to their peculiar barrier characteristics.